A260304 - OEIS

%I #36 Sep 08 2022 08:46:13

%S 2,3,5,10,25,75,250,875,3125,11250,40625,146875,531250,1921875,

%T 6953125,25156250,91015625,329296875,1191406250,4310546875,

%U 15595703125,56425781250,204150390625,738623046875,2672363281250,9668701171875,34981689453125,126564941406250

%N a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.

%C Lim_{n -> infinity} a(n + 1)/a(n) = 2 + phi = 3.6180339887..., where phi is the golden ratio (A001622).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).

%F G.f.: (2 - 7*x)/(1 - 5*x + 5*x^2).

%F a(n) = ((5 + 2*sqrt(5))*((5 - sqrt(5))/2)^n + (5 - 2*sqrt(5))*((5 + sqrt(5))/2)^n)/5.

%F a(n) = 2*A030191(n) - 7*A030191(n-1). - _Bruno Berselli_, Nov 23 2015

%t Table[((5 + 2 Sqrt[5]) ((5 - Sqrt[5])/2)^n + (5 - 2 Sqrt[5]) ((5 + Sqrt[5])/2)^n)/5, {n, 0, 30}]

%t RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 5 a[n - 1] - 5 a[n - 2]}, a, {n, 0, 30}] (* _Bruno Berselli_, Nov 23 2015 *)

%o (Magma) [n le 2 select n+1 else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 23 2015

%o (PARI) a(n)=([0,1; -5,5]^n*[2;3])[1,1] \\ _Charles R Greathouse IV_, Jul 26 2016

%Y Cf. A020876, A030191, A098111.

%Y Cf. A093129: initial values 1,2; A081567: initial values 1,3.

%K nonn,easy

%O 0,1

%A _Ilya Gutkovskiy_, Nov 21 2015

%E Edited by _Bruno Berselli_, Nov 23 2015